High precision positioning systems, such as a microlithographic systems, require a smooth stage motion that produces a minimal amount of structural vibration or oscillation in the system's structure. Although most conventional positioning systems are supported by anti-vibration devices in an attempt to minimize disturbances, the unavoidable acceleration and deceleration of the stage produces forces on the positioning system which often result in small oscillations of the positioning system's structure.
FIGS. 1A-1D show respectively the position, velocity, acceleration, and "jerk" of a positioning system's stage (not shown) moving during a conventional scan. The X axes of FIGS. 1A-1D represent the position, velocity, acceleration, and jerk of the stage with respect to time, which is represented along the Y axes.
As shown in FIG. 1A, the stage moves from position 0 to X between times to and t.sub.0 and t.sub.5. The stage is moving at a constant velocity V.sub.C between times t.sub.2 and t.sub.3 and is stationary before time t.sub.0 and after time t.sub.5 as illustrated in FIG. 1B. However, a settling period exists between time t.sub.2 and t.sub.set, during which any oscillations generated during the acceleration of the stage are allowed to dissipate.
FIG. 1C illustrates the acceleration and deceleration of the stage during a conventional scan. As shown in FIG. 1C, the acceleration increases in a linear manner until at time t.sub.1 the acceleration is at +A, at which time the acceleration of the stage decreases until it is back at 0 at time t.sub.2. Thus, at time t.sub.2 the stage will have a constant velocity V.sub.C shown in FIG. 1B. The stage then linearly decelerates between times t.sub.3 to t.sub.5 with a maximum deceleration at time t.sub.4.
FIG. 1D illustrates the jerk on the stage. The jerk is equal to the derivative of the acceleration of the stage with respect to time. As shown in FIG. 1D, during acceleration the jerk on the stage is at +J from time t.sub.0 to t.sub.0, at -J from time t.sub.1 to t.sub.2, and 0 at time t.sub.2. Thus, as illustrated in FIG. 1D, during acceleration there are discontinuities in the jerk at times t.sub.0, t.sub.1 and t.sub.2. Further, when the acceleration of the stage begins and ends at times t.sub.0 and t.sub.2, the jerk is equal to +J and -J, respectively. A similar, but opposite jerk, occurs during deceleration of the stage, as illustrated in FIG. 1D.
A discontinuous jerk on a stage will create discontinuities in the motion of the stage during the scan, thus vibrating the stage. Moreover, a large jerk at the beginning and end of the acceleration and deceleration of the stage will produce a large reactive force, which consequently excites the positioning system's structure and creates large oscillations. Small vibrations or oscillations in a positioning system, such as a microlithography machine, will have a large deleterious effect where the system is expected to position stages with sub-micron accuracy.
Consequently, in a conventional positioning system in which oscillations occur, a settling period, between times t.sub.2 and t.sub.set, is often required during which the oscillations are allowed to dissipate. The amount of time the stage needs to reach scanning stability at a constant velocity increases due to the increase or need for settling. Thus, the settling period reduces the throughput of a system.